I think it's hard to take anyone seriously when claiming that they have a new model that will revolutionize the foundations of particle physics, molecular biology, metamathematics, economics, evolutionary biology, complexity theory, and others. Of course it's possible, but the magnitude of the claims about vastly disparate fields feels more like someone who only has a bird's-eye view and sees shallow similarities.
Importantly though, neither I nor anyone commenting on Wolfram's plans and thoughts are engaging in mathematics or physics. If anything, the arguments around are mostly to do with the philosophy or sociology of science - looking at how discoveries have usually worked throughout history and comparing that to Wolframs work. To go into the math, you must have some belief that it is worthwhile to spend your time to do quite difficult work, and most arguments around here are about why his work currently doesn't seem to pass that mark. EDIT: to put a number on how difficult engaging with the actual maths and physics would be, according to the article, the Wolfram Research Project has currently published 700 pages worth of discoveries.
To be honest about my own feelings, when I see him claiming his new model will explain what mathematics is - it makes me think he's only a few years away from claiming it explains the meaning of life and why there is something instead of nothing. Perhaps even squaring the circle and computing the number of angels that fit on the head of a pin...